Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 2

For a homogeneous canonical massless scalar field, the flat Friedmann equation and field equation in conformal time arePut $C=\sqrt{3/2}$. On the negative-time branch, $\mathcal{H}=1/(2\tau )$, ${\varphi }_0^{\prime }=C/\tau$ and ${\varphi }_0^{\prime \prime }=-C/{\tau }^2$. ThereforeBoth equations are satisfied. Since $a^{\prime }<0$, this is cosmological contraction with a free homogeneous scalar. Its energy density and pressure are equal, so it is a stiff fluid, and the scale factor tends to zero as $\tau \uparrow 0$.

The massless scalar stress tensor in mixed-index form isIn conformal Newtonian gauge, $g^{00}=-a^{-2}(1-2\Phi )$ and $g^{ij}=a^{-2}(1+2\Psi ){\delta }^{ij}$ to first order. Spatial gradients of the background vanish, so products of two spatial perturbation gradients are second order. ConsequentlySubstitution yieldsThe potential $\Psi$ multiplies only second-order gradient products in these components. The same reasoning makes the off-diagonal spatial stress vanish at first order, consistent with $\Phi =\Psi$.

Let $B=\mathcal{H}\Phi +{\Phi }^{\prime }$. The momentum constraint gives $\delta \varphi =2B/{\varphi }_0^{\prime }$. Differentiating it and using the background field equation givesInsert this into the energy constraint, whose right-hand side in the PDF contains $-{\varphi }_0^{\prime 2}\Phi$:Rearranging and expanding $B$ givesHere ${\varphi }_0^{\prime 2}/2=3{\mathcal{H}}^2$ and ${\mathcal{H}}^{\prime }=-2{\mathcal{H}}^2$. The coefficient of $\Phi$ cancels, provingThis also identifies the derivative lost from the background-field square in the TeX transcription.

For each Fourier mode the equation is ${\Phi }_k^{\prime \prime }+6\mathcal{H}{\Phi }_k^{\prime }+k^2{\Phi }_k=0$. Set ${\Phi }_k=a^{-3}{u}_k$. Direct differentiation removes the first derivative and givesFor $|k\tau |\gg 1$, the inverse-square term is negligible, and the positive incoming-frequency branch has $u_k\sim A_k{e}^{-ik\tau }$. ThereforeThe momentum constraint then giveswhere ${\varphi }_0^{\prime }=-C/a^2$. Thus the scalar perturbation has precisely the required $a^{-1}$ behavior. Conversely, the two constraints imply the potential equation in part (c); reconstructing $\delta \varphi$ by the momentum constraint therefore satisfies the energy constraint too. This checks compatibility beyond simply comparing powers of $a$.

For the exact mode equation write $z=-k\tau >0$ and ${\Phi }_k=(-\tau {)}^{-1}{y}_k(z)$. Substitution givesThe positive incoming-frequency solution is the Newtonian-potential Hankel modes in stiff contraction branchFor large $z$, $H_1^{(1)}(z)\sim \sqrt{2/(\pi z)}{e}^{i(z-3\pi /4)}$. Part (d) and the normalized scalar incoming mode giveAt late times, $H_1^{(1)}(z)\sim -2i/(\pi z)$, so the leading growing cosmological Newtonian potential is $-2i{N}_k/(\pi k{\tau }^2)$. Hence the vacuum-normalized Newtonian spectrum in stiff contraction isFourier-volume conventions can modify the overall normalization but not this exponent. The dimensionless power per logarithmic interval is $k^3⟨|{\Phi }_k{|}^2⟩\propto k^{-1}$, corresponding to $n_s=0$ if written as $k^{n_s-1}$. The question asks the dimensional mode variance's exponent, which is $-4$; it is also important not to substitute a curvature-perturbation spectrum for the specified cosmological Newtonian potential.